#8505332

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#37331	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4416 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

(1,2)(3,7)(4,5)(6,8)(9,10,15,12)(11,16,21,19)(13,22,24,17)(14,18,20,23)(25,26)(27,34)(28,31)(29,32)(30,33)(35,40)(36,38)(37,39);(2,5)(6,7)(9,11)(10,16)(12,19)(13,14)(15,21)(17,23)(18,22)(20,24)(26,28)(29,34)(33,36)(37,40);(1,2)(3,6)(4,5)(7,8)(9,10,15,12)(11,19,21,16)(13,17,24,22)(14,18,20,23)(25,27)(26,29)(28,34)(30,35)(31,32)(33,37)(36,40)(38,39);(1,3,4,8)(2,6,5,7)(9,12,15,10)(11,19,21,16)(13,22,24,17)(14,18,20,23)(25,28,31,26)(27,34,32,29)(30,33,38,36)(35,37,39,40);(9,13)(10,17)(11,20)(12,22)(14,21)(15,24)(16,18)(19,23)(25,29,31,34)(26,32,28,27)(30,37,38,40)(33,39,36,35);(9,14,15,20)(10,18,12,23)(11,13,21,24)(16,22,19,17)(25,30,31,38)(26,33,28,36)(27,35,32,39)(29,37,34,40)

Polycyclic presentation:

2^9|a^2=ghi,c^2=ghi,d^2=hi,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8412528);